It would hover over it like a puck on an air hockey table.
A) If the plane remains stationary relative to the ground, it will not take off.
B) If the plane moves relative to the ground, it will take off.
C) The treadmill by itself can not get the plane to remain stationary relative to the ground, due to lack of traction.
By extension of that, the plane will indeed eventually take off, though it would take a longer treadmill/conveyor belt.
I said air and lift are irrelevant to the argument, because they are. We all know flight is dependent on Bernoulli's Principle, so the original question isn't even very valid. The entire debate is more whether a treadmill is capable of retaining a plane's lateral positioning, which it isn't, nor ever will be.